Static determinacy and stability of a truss

Assumed truss: a rectangular truss with two panels; each panel has a top chord, a bottom chord, vertical members at the ends and a pair of crossing diagonals (cross-bracing). So there are 3 joints on top and 3 at the bottom. The left support is a pin (2 reactions) and the right support is a hinge (2 reactions).

Count

• Joints $j = 6$
• Members: top chords 2 + bottom chords 2 + verticals 3 + diagonals 4 = $m = 11$
• Reactions $r = 2 + 2 = 4$

Static determinacy

For a plane truss, the condition for a determinate truss is $m + r = 2j$.

$$m + r = 11 + 4 = 15,\qquad 2j = 12$$

Since $m + r > 2j$, the truss is statically indeterminate, with degree of indeterminacy

$$D_s = (m + r) - 2j = 15 - 12 = 3$$

Split as external and internal: external degree $= r - 3 = 1$ (the pin plus hinge provide 4 reactions against 3 equilibrium equations), and internal degree $= m - (2j - 3) = 11 - 9 = 2$ (one extra diagonal in each panel).

Stability

• Each panel is a braced rectangle (made of triangles by the diagonals), so the truss is internally stable (rigid), with no mechanism.
• The two supports are both pinned, so the reactions are not all parallel or concurrent and the truss cannot move as a rigid body (no translation or rotation).

Hence the truss is stable but statically indeterminate to the third degree. (If only one diagonal were used in each panel, $m = 9$, $m + r = 13 > 12$ and the truss would be indeterminate to the first degree, externally only.)